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Outranking

ELECTRE I - ELimination Et Choix Traduisant la REalité

Concordance-discordance (crisp outranking)

Roy, B.1968doi:10.1051/ro/196802v100571 ↗

Overview

ELECTRE I produces a partial preorder (outranking graph) and a kernel (non-dominated set), not a complete ranking. An alternative a outranks b if C(a,b) ≥ c̄ AND D(a,b) ≤ d̄. The kernel is the recommended choice set. Alternatives not in the kernel may still be incomparable (not dominated). Adjust thresholds c̄ and d̄ to reflect the decision-maker's tolerance for inconsistency and veto.

Output: preference flow, higher is better
Data: Crisp, complete numeric matrix
Weights: Needs a weight source
Size: 2+ alternatives, 3-12 criteria works best
Used for: Outranking analysis, complex social/economic policy

How it works

1
Euclidean normalisation: transform each criterion column to a dimensionless unit vector. Enables comparison across criteria with different units.
$x_{ij} = \frac{a _{ij}}{\sum _{k = 1}^{m} a _{k j}^{2}}$
Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 Eq.(2-6), p.14
2
Weight the normalised matrix: multiply each column of X by the corresponding criterion weight w_j to obtain the weighted normalised matrix Y.
$Y = X \cdot W, y_{ij} = w_{j} x_{ij}, W = diag (w_{1}, \dots, w_{n}), j \sum w_{j} = 1$
Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 p.15
3
Determine concordance set C_kl and discordance set D_kl for each ordered pair (A_k, A_l): direction-aware via Δ_j(k,l); C_kl contains criteria where A_k is at least as good as A_l; D_kl is the complement.
$Δ_{j} (k, l) = {y_{k j} - y_{l j} y_{l j} - y_{k j} j \in J^{+} j \in J^{-}; C_{k l} = {j : Δ_{j} (k, l) \geq 0}, D_{k l} = {j : Δ_{j} (k, l) < 0}, C_{k l} \cup D_{k l} = {1, \dots, n}$
Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 p.15
4
Construct m×m concordance matrix C (index c_kl = sum of weights in C_kl) and discordance matrix D (index d_kl = max weighted-normalised gap where A_l outperforms A_k, normalised by the global max gap). Diagonal entries undefined.
$c_{k l} = j \in C_{k l} \sum w_{j}, d_{k l} = \frac{max _{j \in D_{k l}} ∣ y _{k j} - y _{l j} ∣}{max _{j} ∣ y _{k j} - y _{l j} ∣}$
Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 Eq.(2-7), p.16
5
Determine concordance dominance matrix F and discordance dominance matrix G using threshold values. Threshold f̄ = average c_kl; threshold d̄ = average d_kl (or user-supplied). F_kl=1 if c_kl ≥ f̄; G_kl=1 if d_kl ≤ d̄ (A_k not strongly opposed on any criterion).
$\overset{ˉ}{f} = \frac{1}{m ( m - 1 )} k \neq = l \sum c_{k l}, F_{k l} = 1 [c_{k l} \geq \overset{ˉ}{f}]; \overset{ˉ}{d} = \frac{1}{m ( m - 1 )} k \neq = l \sum d_{k l}, G_{k l} = 1 [d_{k l} \leq \overset{ˉ}{d}]$
Triantaphyllou 2000 Ch.2 §2.2.5 Eqs.(2-8)(2-9), p.17
6
Compute aggregate dominance matrix E = F ⊗ G (element-wise product). E_kl = 1 means A_k dominates A_l on both concordance and discordance grounds simultaneously.
$E_{k l} = F_{k l} \cdot G_{k l} \in {0, 1}$
Triantaphyllou 2000 Ch.2 §2.2.5 Eq.(2-10), p.17-18
7
Eliminate less favorable alternatives. Any alternative A_l whose column in E contains at least one 1 is dominated. The kernel (choice set) = alternatives with no 1 in their column. If the kernel is a singleton, that is the best alternative; otherwise a partial preorder is obtained.
$Dominated = {A_{l} : \exists k, E_{k l} = 1}, Kernel = A ∖ Dominated$
Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 p.18

Look elsewhere when

•Small dataset (m<3). Outranking benefit minimal.
•Compensatory preferences acceptable. Simpler ranking method.

Assumptions to verify

Decision-maker can specify preference (p), indifference (q), and veto (v) thresholds
Non-compensatory preference structure

Limitations

•Rank reversal known on alternative-set changes (ref: general MCDM literature)

Edge cases and pitfalls

•if c_kl ≥ f̄; G_kl=1 if d_kl ≤ d̄ (A_k not strongly opposed on any criterion).
•If the kernel is a singleton, that is the best alternative; otherwise a partial preorder is obtained.

Treating the kernel as a full ranking: ELECTRE I produces a choice set, not scores. Use ELECTRE II/III/IV for full ranking.
Threshold sensitivity: changing c̄ or d̄ even slightly can alter the kernel substantially - always run sensitivity analysis.
Equal weights assumption: unlike compensatory methods, ELECTRE weights are importance votes, not trade-off rates - use AHP or expert judgment to derive them.

Works with

Commonly takes its weights from

AHP BWM ENTROPY CRITIC SWARA

How to cite

Roy, B. (1968). Classement et choix en présence de points de vue multiples (la méthode ELECTRE). Revue Française d'Informatique et de Recherche Opérationnelle. https://doi.org/10.1051/ro/196802v100571

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

ELECTRE

Overview

Output

preference flow, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Outranking analysis, complex social/economic policy

How it works

Euclidean normalisation: transform each criterion column to a dimensionless unit vector. Enables comparison across criteria with different units.

x_{ij} = \frac{a _{ij}}{\sum _{k = 1}^{m} a _{k j}^{2}}

Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 Eq.(2-6), p.14

Weight the normalised matrix: multiply each column of X by the corresponding criterion weight w_j to obtain the weighted normalised matrix Y.

Y = X \cdot W, y_{ij} = w_{j} x_{ij}, W = diag (w_{1}, \dots, w_{n}), j \sum w_{j} = 1

Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 p.15

Determine concordance set C_kl and discordance set D_kl for each ordered pair (A_k, A_l): direction-aware via Δ_j(k,l); C_kl contains criteria where A_k is at least as good as A_l; D_kl is the complement.

Δ_{j} (k, l) = {y_{k j} - y_{l j} y_{l j} - y_{k j} j \in J^{+} j \in J^{-}; C_{k l} = {j : Δ_{j} (k, l) \geq 0}, D_{k l} = {j : Δ_{j} (k, l) < 0}, C_{k l} \cup D_{k l} = {1, \dots, n}

Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 p.15

Construct m×m concordance matrix C (index c_kl = sum of weights in C_kl) and discordance matrix D (index d_kl = max weighted-normalised gap where A_l outperforms A_k, normalised by the global max gap). Diagonal entries undefined.

c_{k l} = j \in C_{k l} \sum w_{j}, d_{k l} = \frac{max _{j \in D_{k l}} ∣ y _{k j} - y _{l j} ∣}{max _{j} ∣ y _{k j} - y _{l j} ∣}

Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 Eq.(2-7), p.16

Determine concordance dominance matrix F and discordance dominance matrix G using threshold values. Threshold f̄ = average c_kl; threshold d̄ = average d_kl (or user-supplied). F_kl=1 if c_kl ≥ f̄; G_kl=1 if d_kl ≤ d̄ (A_k not strongly opposed on any criterion).

\overset{ˉ}{f} = \frac{1}{m ( m - 1 )} k \neq = l \sum c_{k l}, F_{k l} = 1 [c_{k l} \geq \overset{ˉ}{f}]; \overset{ˉ}{d} = \frac{1}{m ( m - 1 )} k \neq = l \sum d_{k l}, G_{k l} = 1 [d_{k l} \leq \overset{ˉ}{d}]

Triantaphyllou 2000 Ch.2 §2.2.5 Eqs.(2-8)(2-9), p.17

Compute aggregate dominance matrix E = F ⊗ G (element-wise product). E_kl = 1 means A_k dominates A_l on both concordance and discordance grounds simultaneously.

E_{k l} = F_{k l} \cdot G_{k l} \in {0, 1}

Triantaphyllou 2000 Ch.2 §2.2.5 Eq.(2-10), p.17-18

Eliminate less favorable alternatives. Any alternative A_l whose column in E contains at least one 1 is dominated. The kernel (choice set) = alternatives with no 1 in their column. If the kernel is a singleton, that is the best alternative; otherwise a partial preorder is obtained.

Dominated = {A_{l} : \exists k, E_{k l} = 1}, Kernel = A ∖ Dominated

Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 p.18

Edge cases and pitfalls

•if c_kl ≥ f̄; G_kl=1 if d_kl ≤ d̄ (A_k not strongly opposed on any criterion).

•If the kernel is a singleton, that is the best alternative; otherwise a partial preorder is obtained.

Treating the kernel as a full ranking: ELECTRE I produces a choice set, not scores. Use ELECTRE II/III/IV for full ranking.

Threshold sensitivity: changing c̄ or d̄ even slightly can alter the kernel substantially - always run sensitivity analysis.

Equal weights assumption: unlike compensatory methods, ELECTRE weights are importance votes, not trade-off rates - use AHP or expert judgment to derive them.